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  • distance of non-parallel lines | planetmath.org
    derive the expression of the distance d d d between two non parallel straight lines in ℝ 3 superscript ℝ 3 mathbb R 3 Suppose that the position vectors of the points of the two non parallel lines are expressed in parametric forms r a s u normal r normal a s normal u vec r vec a s vec u and r b t v normal r normal b t normal v vec r vec b t vec v where s s s and t t t are parameters A common normal vector of the lines is the cross product u v normal u normal v vec u times vec v of the direction vectors of the lines and it may be normed to a unit vector n u v u v assign normal n normal u normal v normal u normal v vec n frac vec u times vec v vec u times vec v by dividing it by its length which is distinct from 0 because of the non parallelity The vectors a normal a vec a and b normal b vec b are the position vectors of certain points A A A and B B B on the lines and thus their difference a b normal a normal b vec a vec b is the vector from B B B to A A A If we project a b normal a normal b vec a vec b on the unit normal n normal n vec n the obtained vector d a b n n assign normal d normal normal a normal b normal n normal n vec d vec a vec b cdot vec n vec n has the sought length d a b n d normal normal a normal b normal n d vec a

    Original URL path: http://www.planetmath.org/distanceofnonparallellines (2016-04-25)
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  • divisibility by prime number | planetmath.org
    2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A15A72 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit divisibility by prime number Theorem Let a a a and b b b be integers and p p p any prime number Then we have p a b p a p b fragments p normal a b normal p normal a p normal b displaystyle p mid ab quad Leftrightarrow quad p mid a lor p mid b 1 Proof Suppose that p a b fragments p normal a b p mid ab Then either p a fragments p normal a p mid a or p a not divides p a p nmid a In the latter case we have gcd a p 1 a p 1 gcd a p 1 and therefore the corollary of Bézout s lemma gives the result p b fragments p normal b p mid b Conversely if p a fragments p normal a p mid a or p b fragments p normal b p mid b then for example a m p a m p a mp for some integer m m m this implies that a b m b p a b normal m b p ab mb cdot p i e p a b fragments p normal a b p mid ab Remark 1 The theorem means that if a product is divisible by a prime number then at least one of

    Original URL path: http://www.planetmath.org/divisibilitybyprimenumber (2016-04-25)
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  • divisibility by product | planetmath.org
    3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A11A05 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit divisibility by product Theorem Let R R R be a Bézout ring i e a commutative ring with non zero unity where every finitely generated ideal is a principal ideal If a b c a b c a b c are three elements of R R R such that a a a and b b b divide c c c and gcd a b 1 a b 1 gcd a b 1 then also a b a b ab divides c c c Proof The divisibility assumptions mean that c a a 1 b b 1 c a subscript a 1 b subscript b 1 c aa 1 bb 1 where a 1 subscript a 1 a 1 and b 1 subscript b 1 b 1 are some elements of R R R Because R R R is a Bézout ring there exist such elements x x x and y y y of R R R that gcd a b 1 x a y b a b 1 x a y b gcd a b 1 xa yb This implies the equation a 1 x a a 1 y b a 1 x b b 1 y b a 1 subscript a 1 x a subscript a 1 y b subscript a 1 x b subscript b 1 y b subscript a 1 a 1 xaa 1 yba 1 xbb 1 yba 1 which shows that a 1 subscript a 1 a

    Original URL path: http://www.planetmath.org/divisibilitybyproduct (2016-04-25)
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  • divisibility in rings | planetmath.org
    2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A13A05 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit Let A A normal A cdot be a commutative ring with a non zero unity 1 If a a a and b b b are two elements of A A A and if there is an element q q q of A A A such that b q a b q a b qa then b b b is said to be divisible by a a a it may be denoted by a b fragments a normal b a mid b If A A A has no zero divisors and a 0 a 0 a neq 0 then q q q is uniquely determined When b b b is divisible by a a a a a a is said to be a divisor or factor of b b b On the other hand b b b is not said to be a multiple of a a a except in the case that A A A is the ring ℤ ℤ mathbb Z of the integers In some languages e g in the Finnish b b b has a name which could be approximately be translated as containant b b b is a containant of a a a b b b on a a a n sisältäjä Properties a b fragments a normal b a mid b iff b a b a b subseteq a see the principal ideals Divisibility is a reflexive and transitive relation in A A A 0 is divisible by all elements of A A A a 1 fragments a normal 1 a mid 1 iff a a a is a unit of A A A All elements of A A A are divisible by every unit of A A A If a b fragments a normal b a mid b then a n b n n 1 2 fragments superscript a n normal superscript b n fragments normal n 1 normal 2 normal normal normal a n mid b

    Original URL path: http://www.planetmath.org/divisibilityinrings (2016-04-25)
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  • divisibility of nine-numbers | planetmath.org
    http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A11A51 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A11A51 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit divisibility of nine numbers We know that 9 is divisible by the prime number 3 and that 99 by another prime number 11 If we study the divisibility other nine numbers by primes we can see that 999 is divisible by a greater prime number 37 and 9999 by 101 which also is a prime and so on Such observations may be generalised to the following Proposition For every positive odd prime p p p except 5 there is a nine number 999 9 999 9 999 9 divisible by p p p Proof Let p p p be a positive odd prime 5 absent 5 neq 5 Let s form the set of the integers 9 99 999 99 9 p nines 9 99 999 normal subscript normal 99 9 p nines displaystyle 9 99 999 ldots underbrace 99 9 p mathrm nines 1 We make the antithesis that no one of these numbers is divisible by p p p Therefore their least nonnegative remainders modulo p p p are some of the p 1 p 1 p 1 numbers 1 2 3 p 1 1 2 3 normal p 1 displaystyle 1 2 3 ldots p 1 2 Thus there are at least two of the numbers 1 say a a a and b b b a b a b a b having the same remainder The difference b a b a b a then

    Original URL path: http://www.planetmath.org/divisibilityofninenumbers (2016-04-25)
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  • division | planetmath.org
    PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A11A05 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit Division is the operation which assigns to every two numbers or more generally elements of a field a a a and b b b their quotient or ratio provided that the latter b b b is distinct from zero The quotient or ratio a b a b frac a b of a a a and b b b may be defined as such a number or element of the field x x x that b x a normal b x a b cdot x a Thus b a b a normal b a b a b cdot frac a b a which is the fundamental property of quotient The quotient of the numbers a a a and b b b 0 absent 0 neq 0 is a uniquely determined number since if one had a b x y a b a b x y a b frac a b x neq y frac a b then we could write b x y b x b y a a 0 b x y b x b y a a 0 b x y bx by a a 0 from which the supposition b 0 b 0 b neq 0 would imply x y 0 x y 0 x y 0 i e x y x y x y The explicit general expression for a b a b frac a b is a b b 1 a a b normal superscript b 1 a frac a b b 1 cdot a where b 1 superscript b 1 b 1 is the inverse number the multiplicative inverse of a a a because b b 1 a b b 1 a 1 a a b superscript b 1 a b superscript b 1 a 1 a a b b 1 a bb 1 a 1a a For positive numbers the quotient may be obtained by performing the division algorithm with a a a and b b b If a b 0 a b 0 a b 0 then a b a b frac a b indicates how many times b b b fits in a a a The quotient of a a a and b b b does not change if both numbers elements are multiplied or divided which action is called reduction by any k 0 k 0 k neq 0 k a k b k b 1 k a b 1 k 1 k a b 1 a a b k a k b superscript k b 1 k a superscript b 1 superscript k 1 k a superscript

    Original URL path: http://www.planetmath.org/division (2016-04-25)
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  • division in group | planetmath.org
    sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A12E99 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit division in group In any group G G G cdot one can introduce a division operation by setting x y x y 1 normal x y normal x superscript y 1 x y x cdot y 1 for all elements x x x y y y of G G G On the contrary the group operation and the unary inverse forming operation may be expressed via the division by x y x y y y x 1 x x x fragments x normal y x normal fragments normal fragments normal y normal y normal normal y normal normal superscript x 1 fragments normal x normal x normal normal x normal displaystyle x cdot y x y y y quad x 1 x x x 1 The division which of course is not associative has the properties 1 x z y z x y fragments fragments normal x normal z normal normal fragments normal y normal z normal x normal y normal x z y z x y 2 x y y x fragments x normal fragments normal y normal y normal x normal x y y x 3 x x y z z y fragments fragments normal x normal x normal normal fragments normal y normal z normal z normal y normal x x y z z y The above result may be conversed Theorem If the operation of the non empty groupoid G G G has the properties 1 2 and 3 then G G G equipped with the multiplication and inverse forming by 1 is a group Proof Here we prove only the associativity of normal cdot First we derive some auxiliary results Using definitions and the properties 1 and 2 we obtain x y y 1 x y y y y x y y x fragments fragments normal x normal y normal normal superscript y 1 fragments normal x normal y normal normal fragments normal fragments normal y normal y normal normal y normal x normal fragments normal y normal y normal x normal x y y 1 x y y y

    Original URL path: http://www.planetmath.org/divisioningroup (2016-04-25)
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  • divisor as factor of principal divisor | planetmath.org
    22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error Socket error Could not connect to http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A20 00 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A20 00 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit divisor as factor of principal divisor Let an integral domain mathcal O have a divisor theory normal superscript mathcal O to mathfrak D The definition of divisor theory implies that for any divisor mathfrak a there exists an element ω ω omega of mathcal O such that mathfrak a divides the principal divisor ω ω omega i e that ω ω mathfrak ac omega with mathfrak c a divisor The following theorem states that mathfrak c may always be chosen such that it is coprime with any beforehand given divisor Theorem For any two divisors mathfrak a and mathfrak b there is a principal divisor ω ω omega such that ω ω mathfrak ac omega and gcd 1 1 gcd mathfrak b mathfrak c 1 Proof Let 1 s subscript 1 normal subscript s mathfrak p 1 ldots mathfrak p s all distinct prime divisors which divide the product mathfrak ab and let the divisor mathfrak a be exactly divisible by the powers 1 a 1 s a s superscript subscript 1 subscript a 1 normal superscript subscript s subscript a s mathfrak p 1 a 1 ldots mathfrak p s a s the cases a i 0 subscript a i 0 a i 0

    Original URL path: http://www.planetmath.org/divisorasfactorofprincipaldivisor (2016-04-25)
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